Nonautonomous Discrete Dynamics: Discretization and Taylor Approximation

In this talk, we consider a class of nonautonomous difference equations (recursions) possessing an attractive invariant manifold in the extended state space. It is our aim to study the behavior of this manifold, as well as of its corresponding asymptotic phase under perturbation. Our framework is sufficiently general to include discrete counterparts of (center-)unstable and inertial manifolds. Two main applications of such results for so-called invariant fiber bundles will be discussed: • They indicate that attractive invariant manifolds of evolutionary equations persist under a large class of numerical discretizations. Moreover, the rate of convergence can be estimated. Similar results hold for the associated asymptotic phase. • We develop a method to calculate their Taylor approximation. Here, the desired Taylor coefficients are determined by bounded solutions of a certain linear difference equation in the space of multilinear mappings, instead of an algebraic equation occurring in the classical autonomous theory. Even in finite dimensions, such a technique is of crucial importance for an application of the reduction principle in a nonautonomous stability theory.